Finite element method: the basics

Finite element method: the basicsProf. José E. Andrade

Department of Civil & Environmental Engineering

July, 2007

PLAXIS FINITE ELEMENT CODE FOR SOIL AND ROCK ANALYSES

Plaxis BVPO Box 572

2600 AN DelftThe Netherlands

Tel: +31 (0)15 251 77 20Fax: +31 (0)15 257 31 07

E-Mail: info@plaxis.nlWebsite: www.plaxis.nl

19 - 21 March 2007 Course Computational Geotechnics (German)Stuttgart, Germany

26 - 29 March 2007International course for experienced Plaxis usersAntwerpen, Belgium

1 - 4 April5th International Workshop on Applicationsof Computational Mechanics in Geotechnical EngineeringGuimaraes, Portugal

18 - 19 April 2007Plaxis user meeting UKLondon, Manchester, United Kingdom

25 - 27 April 2007NUMOG XTenth International Symposium on Numerical Models in GeomechanicsRhodes, Greece

5 - 10 May 2007ITA-Aites World Tunnel Congress 2007Prague, Czech Republic

8 - 11 May 200716th Southeast Asian Geotechnical ConferenceSelangor Darul Ehsan, Malaysia

12 - 14 June 2007 Course Computational GeotechnicsManchester, United Kingdom

25 - 28 June 2007Course on advanced Computational GeotechnicsSydney, Australia

16 - 20 July 2007XIII PCSMGE 2007Isla de Margarita, Venezuela

24 - 27 July 2007Course on Computational Geotechnics and DynamicsChicago, USA

10 - 14 September 2007Course on advanced Computational GeotechnicsAnkara, Turkey

24 - 27 September 2007XIV ECSMGE Madrid, Spain

21 - 24 October 200710 ANZ SMGE,Brisbane, Australia

7 - 9 November 200714th European Plaxis User MeetingKarlsruhe, Germany

26 - 28 November 200714 African Regional Conference SMGE, Yaounde, Cameroon-Africa

10 - 14 December 200713th Asian Regional Conference on Soil Me-chanics and Geotechnical EngineeringCalcutta, India

Activities 2007

7005

910

Outline

• Applications of FEM

• Fundamentals

• Examples

Applications of FEM

Biomechanics

bypass alternatives

patient-specificmodel

Geomechanics

COMPACTIVEZONE

DILATIVEZONE

!a

r

GRAIN

VOID

SHEARBAND

P

FIELD SCALE SPECIMEN SCALE

'HOMOGENEOUS'

SOIL

MESO SCALE GRAIN SCALE

-3 -2 -1 0 >1LOG (m)

FOOTING

FAILURE

SURFACE!

Solid-fluid interactions

DEVIATORIC STRAIN

0 .0 5

0 .1

0 .1 5

0 .2

0 .2 5

0 .3

shear strain

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

PRESSURE, kPa

VERT

ICA

L C

OO

RDIN

ATE

, m

FE SOLNANAL SOLN

consolidation

Simulation engineering

Fundamentals of FEM

FEM

• Designed to approximately solve PDE’s

• PDE’s model physical phenomena

• Three types of PDE’s:

• Parabolic: fluid flow

• Hyperbolic: wave eqn

• Elliptic: elastostatics

!!"!#$

%!"!#$

&

&

' '() '(* '(+ '(, '(- '(. '(/ '(0 '(1 )'

'()

'(*

'(+

'(,

'(-

'(.

'(/

'(0

'(1

)

'

'()

'(*

'(+

'(,

'(-

'(.

'(/

'(0

'(1

)

FEM recipe

Strong from

Weak form

Galerkin form

Matrix form

Elastostatics: strong

PDE

B.C.’s

u,xx +f = 0u(1) = g

u,x (0) = h

strong form

derived from continuum mechanicsstrong form = PDE + B.C.’s

usually, there is no exact sln for strong form

x10

Elastostatics: weak

•Use principle of virtual work•Introduce virtual displacement•Use strong form∫ 1

0

w(u,xx +f) dx = 0

w; w(1) = 0

∫ 1

0

w,x u,x dx =

∫ 1

0

wf dx + w(0)h

Elastostatics: Galerkin

Construct approximate solution uh

= vh

+ gh

like w

like g

Construct functions based on shape functions

wh

=

n∑

A=1

NAcA vh

=

n∑

A=1

NAdA

gh= gNn+1

Piecewise linear FE

x

10

1

N1

Nn+1

NA

xn+1

xn

x1

x2

xA-1

xAxA+1

finite element

node

Elastostatics: matrixUse weak form, plug-in Galerkin approximation

n∑

B=1

KABdB = FA KAB =

∫ 1

0

NA,xNb,x dx

FA =

∫ 1

0

NAf dx + NA(0)h −

∫ 1

0

NA,xNn+1,x dxg

it all boils down to...

stiffness matrix

K · d = F

force vector

Properties of K · d = F

•Stiffness matrix is•symmetric•banded•positive-definite

•Displacement vector = unknowns only•Can use any linear algebra solver to find solution

may not alwaysapply

Multi-D deformation

!g

!h

!

∇ · σ + f = 0 in !

u = g on "g

σ · n = h on "h

equilibriume.g., clampe.g., confinement

Constitutive relation ugiven get !

e.g., elasticity, plasticity

FEM programTIME STEP LOOP

ITERATION LOOP

ASSEMBLE FORCE VECTOR

AND STIFFNESS MATRIX

ELEMENT LOOP: N=1, NUMEL

GAUSS INTEGRATION LOOP: L=1, NINT

CALL MATERIAL SUBROUTINE

CONTINUE

CONTINUE

CONTINUE

T = T + !T

Element technology: 2D

Serendipity family of quads

Lagrange family of quads

Standard triangularelements

displacement nodeGauss integration point

Modeling ingredients

1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve

4.417

B

H

Modeling ingredients

1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve

1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve

Modeling ingredients

1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve

1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve

Modeling ingredients

1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve

1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve

!a

!r

Modeling ingredients

1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve

1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve

!a

!r

Examples

Hyperbolic: LSST array

output

input

Geometry and B.C.s

input

output

Material parameters

shear modulus degradation & damping

Input base acceleration

Acceleration output

Soil-fluid interaction

• Nonlinear continuum mechanics

• Robust constitutive theory

• Computational inelasticity

• Nonlinear finite elements

Model ingredients

!s

"#

"

x1

x2

X

x

!f

"#

f

Soil-fluid interaction

• Nonlinear continuum mechanics

• Robust constitutive theory

• Computational inelasticity

• Nonlinear finite elements

Model ingredients

! =’"

! !’ = ’2 3

Soil-fluid interaction

• Nonlinear continuum mechanics

• Robust constitutive theory

• Computational inelasticity

• Nonlinear finite elements

Model ingredients

!"

!#

!$

!n

!n+1

!n+1

tr

Fn+1

Fn

Soil-fluid interaction

• Nonlinear continuum mechanics

• Robust constitutive theory

• Computational inelasticity

• Nonlinear finite elements

Model ingredients

Pressure nodeDisplacement node

Plane-strain compressspecific volume

CT scan FE model

Plane-strain compressspecific volume

shear strain and flow fluid pressure

Plane-strain compress

Questions?